Random World: Book - e The Story of a Number

$\mathbf{y = e^x}$ The function that equals its own derivative

1. The derivative of an exponential function is proportional to the function itself

Graph of increasing exponential function

2. In applications one finds numerous phenomena in which rate of change of change of some quantity is proportional to the quantity itself.

3. Any such phenomenon is governed by the differential equation dy/dx = ay, where the constant a determines the rate of change in each case. The solution is $y = ce^a^x$ , where the arbitrary constant C is determined from the initial condition of the system: the value of y when x = 0

Family of exponential curves

4. $y = ce^a^x$ Depending on whether whether a is positive or negative, y will increase or decrease with x, resulting in an exponential growth or decay.

Examples

a. The rate of decay of a radioactive substance-and the amount of radiation it emits, is at every moment proportional to its mass m: $\frac{\mathrm{d}m }{\mathrm{d} t} = -am => m = m_0e^-a^t$

The value of a determines the rate of decay of the substance and is usually measured by the half-life time, the time it takes a radioactive substance to decay to one-half of its initial mass.

Different substances have vastly different half-life times. For example, the common isotope of uranium (U238) has a half-life of about five billion years, ordinary radium (Ra226) about sixteen hundred years, while Ra220 has a half-life of only twenty-three milliseconds.

This explains why some of the unstable elements in the periodic table are not found in natural minerals: whatever quantity may have been present when the earth was born has long since been transformed into more stable elements.

b. When a hot object at temperature To is put in an environment of temperature T, (itself assumed to remain constant), the object cools at a rate proportional to the difference T - T, between its temperature at time t and the surrounding temperature. This is Newton's law of cooling

$\frac{\mathrm{d}T }{\mathrm{d} t} = -a (T-T_1) => T = T_1 + (T_0 - T_1) \; e-^a^t$

c. When sound waves travel through air (or any other medium), their intensity is governed by the differential equation dI/dx = -aI, where x is the distance traveled. The solution, $\frac{\mathrm{d}I }{\mathrm{d} x} = -a I => I = I_0 e - ^a^x$

d. A frequent problem in mechanics is that of describing the motion of a vibrating system-a mass attached to a spring, for example taking into account the resistance of the surrounding medium. This
problem leads to a second-order differential equation with constant coefficients.

An example of such an equation is $\frac{\mathrm{d}^2 y}{\mathrm{d} t^2} + 5 \frac{\mathrm{d} y}{\mathrm{d} t} + 6y = 0$ . The solution is of the form $y = A e^m^t$ where A and m are as yet undetermined constants.

We have two distinct solutions $y = A e-a^2^t$ and $y = A e-a^3^t$ . Substituting we get the equation $y = A e-^2^t + \: Be-^3^t$ is also a solution.

Book - e The Story of a Number - Eli Maor